The point which does not lie in the solution region of 2x-y≤ 4 is

The correct answer is (5, 1).

To determine which point does not lie in the solution region of the inequality, we can substitute the coordinates of each point $(x,y)$ into the given inequality:
$2x-y\le 4$

If the substitution results in a true statement, then the point lies in the solution region. If it results in a false statement, the point does not lie in the solution region.

Let us test the given point (5, 1):
Here, we have $x=5$ and $y=1$.
Substitute these values into the left-hand side of the inequality:
$2\left(5\right)-1=10-1=9$

Now, compare this result with the right-hand side of the inequality:
$9\le 4$

Since 9 is greater than 4, this statement is false. Therefore, the point (5, 1) does not satisfy the inequality, meaning it does not lie in the solution region.

Let us verify the other points to ensure they satisfy the inequality:

1. For the point (1, -1):
$2\left(1\right)-(-1)=2+1=3$
Since $3\le 4$ is true, this point lies in the solution region.

2. For the point (0, 2):
$2\left(0\right)-2=0-2=-2$
Since $-2\le 4$ is true, this point lies in the solution region.

3. For the point (1, 1):
$2\left(1\right)-1=2-1=1$
Since $1\le 4$ is true, this point lies in the solution region.

Thus, only the point (5, 1) does not lie in the solution region.